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Mirrors > Home > MPE Home > Th. List > Mathboxes > aaanv | Structured version Visualization version GIF version |
Description: Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2353. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
aaanv | ⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1915 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | aaan 2353 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
4 | 3 | bicomi 226 | 1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
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